Now consider the \(UVW\)  outer bound in [1]}.
For any \((R_{0},R_{1}, R_{2}) \in \mathcal {C}(S) \)  and \(0 <\mu _{1} \le \mu _{2}\le 1\) , we have
\(& R_{0} + \mu _{2}R_{2} + \mu _{1} R_{1} \nonumber \\&\le (1-\mu _{2})R_{0} + (\mu _{2} -\mu _{1})(R_{0}+R_{2}) + \mu _{1} (R_{0}+R_{2} +R_{1}) \\&\le \min \lbrace I(U;{Y}_{1}), I(U; {Y}_{2})\rbrace  + \mu _{2} I(V; {Y}_{2} | U) + \mu _{1}I({X}; {Y}_{1} | V, U) \\& \le \lambda I(U;{Y}_{2}) + (1-\lambda ) I(U; {Y}_{1}) + \mu _{2} I(V; {Y}_{2} | U) + \mu _{1}I({X}; {Y}_{1} | V, U) \\&= \mu _{1}h({X}+{N}_{1} | U,V) - \mu _{2} h({X}+{N}_{2} | U,V) + (\mu _{2}- \lambda ) h({X}+{N}_{2} | U) - (1-\lambda ) h({X}+{N}_{1} | U) \nonumber \\&\quad - \mu _{1}h({N}_{1}) + \lambda h({X} + {N}_{2}) + (1-\lambda ) h({X}+{N}_{1}) \\&\le \mu _{1}h({X}+{N}_{1} | U,V) - \mu _{2} h({X}+{N}_{2} | U,V) + (\mu _{2}- \lambda ) h({X}+{N}_{2} | U) - (1-\lambda ) h({X}+{N}_{1} | U) \nonumber \\&\quad - \frac{\mu _{1}}{2} \log \left| (2\pi e) K_{1} \right| + \frac{\lambda }{2} \log \left| (2\pi e) (S + K_{2}) \right| + \frac{1-\lambda }{2} \log \left| (2\pi e) (S + K_{1}) \right|. \) 
